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126                      ANALYTICAL MECHANICS
normal kinetic reaction and is oppositely directed. Therefore the reaction of the surface is directed towards the center of the circular path.
Equations (1) and (2) are independent of the special method used to keep the particle in a circular path. If, for instance, the particle were connected to the center by means of an inextensible string and then projected in a direction perpendicular to the string the results would have been the same.
The force which constrains the particle to move in the circular path is called the central force. This force may be the reaction of a surface, the tension of a string, or the pull of a center of attraction. In order to emphasize the fact that this force is directed towards the center it is often denoted by Fc. Since the subscript makes clear the fact that the force is directed towards the center, we can drop the negative sign from equation (2), and write
F                                              (3)
where P is the time of one revolution.
1.   A particle of mass mi, which describes a circle on a perfectly smooth horizontal table, is connected with another particle of mass m2 which hangs freely; the string which connects the two particles passes through a smooth hole in the table.   Find the condition necessary to keep m2 at rest.
2.   Find the smallest horizontal velocity with which a body must be projected at the equator in order that the body may become a satellite. Find the period of revolution.
3.   A locomotive weighing 125 tons moves in a curve of 600 feet radius, with a velocity of 20 miles per hour.   Find the lateral pressure on the rails if they are on the same level.
4.   Derive the expression for the period of a conical pendulum.
5.   A number of particles of different masses are suspended from the same point by means of strings of different lengths.   Show that when